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Fractions: Computations & Operations Teaching for Conceptual Understanding

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Introduction Represent the following operation using each of the representations:Represent the following operation using each of the representations: Describe what the solution means in terms of each representation you use.Describe what the solution means in terms of each representation you use. PicturesPictures ManipulativesManipulatives Real World SituationsReal World Situations SymbolicSymbolic Oral/Written LanguageOral/Written Language

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Fractional – Parts: Counting 5 Fourths 3 Fourths 10 Fourths 10 Twelfths

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Discussion What does the bottom number in a fraction tell us?What does the bottom number in a fraction tell us? What does the top number in a fraction tell us?What does the top number in a fraction tell us? Misleading notion of fractions: Top number tells “how many.” Bottom number tells “how many parts to make a whole.”

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For example: If a pizza is cut in 12 pieces, 2 pieces makes 1/6 of the pizza –If a pizza is cut in 12 pieces, 2 pieces makes 1/6 of the pizza – Here, the bottom number does not tell us how many parts make up the whole!Here, the bottom number does not tell us how many parts make up the whole! Better Idea! We can assume the top number “counts” while the bottom number tells “what is being counted.” -¾ is a count of three things called fourths Using this notion, what is another way to say/write Thirteen sixths? Explain your rationale.

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Activity: Using the manipulatives, your task is to find a single fraction that names the same amount as:Using the manipulatives, your task is to find a single fraction that names the same amount as: – must be able to provide an explanation for your result. Then, determine the mixed number for 17/4 and provide a justification for your result.Then, determine the mixed number for 17/4 and provide a justification for your result.

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Parts & Whole Tasks: If 8 counters are a whole set, how many are in one-fourth of a set? If 15 counters are a whole, how many counters make three- fifths? If 9 counters are a whole, how many are in five-thirds of a set?

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If 12 counters are ¾ of a set, how many counters are in the full set? If 10 counters are five-halves of a set, how many counters are in one set? (What must be half of one set?) Dark Green Yellow Purple If purple is 1/3, what strip is the whole? If dark green is 2/3, what strip is the whole? If yellow is 5/4, what strip is 1 whole?

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Dark Green Yellow Dark Green Blue If the dark green is the whole, what fraction is the yellow strip? If the dark green strip is one whole, what fraction is the blue strip?

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Getting to Conceptual Understanding: “A father has left his three sons 35 camels to divide among them in this way:“A father has left his three sons 35 camels to divide among them in this way: One-half to one brother, one-third to another brother, and one-ninth to the third brother.” How many camels does each brother receive? Explain your solution to this problem.

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Benchmarks of Zero, One-Half, and One Sort the following fractions into three groups: close to zero, close to ½, and close to one. Provide rationale and explanation for your choices. Close Fractions: Name a fraction that is close to one but not more than one. Name another fraction that is even closer to one. Explain why you believe this fraction is even closer to one than the first. Show using manipulatives.

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Partner/ Group Activity:

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Exploring Ordering Fractions: Exploring Ordering Fractions: Task: Which fraction in each pair is greater? Give an explanation for why you think so?Task: Which fraction in each pair is greater? Give an explanation for why you think so? What are some of the usual approaches that students would choose to use in this activity? Why do you think this is?What are some of the usual approaches that students would choose to use in this activity? Why do you think this is?

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Conceptual Thought Patterns More of the same size parts – to compare 3/8 and 5/8More of the same size parts – to compare 3/8 and 5/8 - students will choose 5/8 as larger because 5>3; right choice, wrong reason;- students will choose 5/8 as larger because 5>3; right choice, wrong reason; - comparing 3/8 and 5/8 should be like comparing 3 apples and 5 apples;- comparing 3/8 and 5/8 should be like comparing 3 apples and 5 apples;

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Same number of parts, but parts of different sizesSame number of parts, but parts of different sizes – to compare ¾ and 3/7 Misconception – students will choose 3/7 as the larger because 7 is more than 4 and the top numbers are the same;Misconception – students will choose 3/7 as the larger because 7 is more than 4 and the top numbers are the same; However, if a whole is divided into 7 parts, the parts will be smaller than if divided into only 4 parts; thus, ¾ is largerHowever, if a whole is divided into 7 parts, the parts will be smaller than if divided into only 4 parts; thus, ¾ is larger Like comparing 3 apples with 3 melons – same number of things but melons are bigger;Like comparing 3 apples with 3 melons – same number of things but melons are bigger;

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More or less than one-half or one wholeMore or less than one-half or one whole – comparing 3/7 and 5/8 – comparing 3/7 and 5/8 3/7 is less than half of the number of sevenths needed to make a whole, so 3/7 is less than one-half;3/7 is less than half of the number of sevenths needed to make a whole, so 3/7 is less than one-half; thus, 5/8 is more than one-half and is therefore the larger fraction;thus, 5/8 is more than one-half and is therefore the larger fraction;

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Distance from one-half or one wholeDistance from one-half or one whole – compare 9/10 and ¾ Misconception – 9/10 is bigger because 9 and 10 are the bigger numbersMisconception – 9/10 is bigger because 9 and 10 are the bigger numbers 9/10 is larger than ¾ because although each is one fractional part away from a whole, tenths are smaller than fourths and so 9/10 is closer to one whole.9/10 is larger than ¾ because although each is one fractional part away from a whole, tenths are smaller than fourths and so 9/10 is closer to one whole.

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Addition and Subtraction – Explorations : “Paul and his brother were each eating the same kind of candy bar. Paul had ¾ of his candy bar. His brother had 7/8 of a candy bar. How much candy did the two boys have together?”“Paul and his brother were each eating the same kind of candy bar. Paul had ¾ of his candy bar. His brother had 7/8 of a candy bar. How much candy did the two boys have together?” Using drawings &/or manipulatives, solve this problem without setting it up in the usual manner and finding common denominators?Using drawings &/or manipulatives, solve this problem without setting it up in the usual manner and finding common denominators? Can you think of two different methods?Can you think of two different methods?

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One possible Solution: We could take a fourth from the 7/8 and add it to the ¾ to make a whole. That would leave 5/8. Thus, the total eaten would be 1 5/8 of candy bar.

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Using Cuisenaire Rods for Fraction Computation Jack and Jill ordered two identical sized pizzas, one cheese and one pepperoni. Jack ate 5/6 of a pizza and Jill ate ½ of a pizza. How much pizza did they eat together?Jack and Jill ordered two identical sized pizzas, one cheese and one pepperoni. Jack ate 5/6 of a pizza and Jill ate ½ of a pizza. How much pizza did they eat together? What would we expect our students to show that would demonstrate their conceptual understanding?What would we expect our students to show that would demonstrate their conceptual understanding?

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Solution Find a strip for the whole that allows both fractions to be modeled.Find a strip for the whole that allows both fractions to be modeled. Dark Green Light Green Dark Green YellowLight Green Red Yellow Whole 5/6 1/2 A Red is 1/3 of the Dark Green, so the solution is 1 1/3.

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Multiplication of Fractions: Beginning Concepts Using visual representations/manipulatives, model the following computation:Using visual representations/manipulatives, model the following computation: “There are 15 cars in Michael’s toy car collection. Two- thirds of the cars are red. How many red cars does Michael have?”

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Using visual representations/manipulatives, now model these following problems:Using visual representations/manipulatives, now model these following problems: You have ¾ of a pizza left. If you give 1/3 of the left- over pizza to your brother, how much of the whole pizza will your brother get?You have ¾ of a pizza left. If you give 1/3 of the left- over pizza to your brother, how much of the whole pizza will your brother get? Someone ate 1/10 of the cake, leaving only 9/10. If you eat 2/3 of the cake that is left, how much of the whole cake have you eaten?Someone ate 1/10 of the cake, leaving only 9/10. If you eat 2/3 of the cake that is left, how much of the whole cake have you eaten? Gloria used 2 ½ tubes of blue paint to paint the sky in her picture. Each tube holds 4/5 ounce of paint. How many ounces of blue paint did Gloria use?Gloria used 2 ½ tubes of blue paint to paint the sky in her picture. Each tube holds 4/5 ounce of paint. How many ounces of blue paint did Gloria use?

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When pieces must be subdivided into smaller unit parts, the problems become more challenging: Zack had 2/3 of the lawn left to cut. After lunch he cut ¾ of the grass that he had left. How much of the whole lawn did he cut after lunch?Zack had 2/3 of the lawn left to cut. After lunch he cut ¾ of the grass that he had left. How much of the whole lawn did he cut after lunch? Bill drank 1/5 of his pop before lunch. After lunch he drank 2/3 of what was left. How much pop did he drink after lunch?Bill drank 1/5 of his pop before lunch. After lunch he drank 2/3 of what was left. How much pop did he drink after lunch?

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Division of Fractions: Beginning Concepts Think about the following problem:Think about the following problem: “Cassie has 5 ¼ yards of ribbon to make three bows for birthday packages. How much ribbon should she use for each bow if she wants to use the same length of ribbon for each?” -What types of solutions would we anticipate our students to come up with? -How could we model the solution using manipulatives/multiple representations? -How many different ways?

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In the following problem, the parts must be split into smaller parts:In the following problem, the parts must be split into smaller parts: “Mark has 1 ¼ hours to finish his three household chores. If he divides his time evenly, how many hours can he give to each?” 1 ¼ Hours 5/12 hour per chore

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Questions/Discussion Inverse vs. ReciprocalInverse vs. Reciprocal 4/5 - representations:4/5 - representations: Think of the many forms that even the symbolic can be represented – decimals, rates, ratios, etc. Other? More???Other? More???

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Applying this Reasoning: Solving Problems

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[A] A student is sorting into stacks a room full of food donated by the school for the local food bank. He sorted of it before lunch and then sorted of the remainder before school ended. What part (fraction) of all the food will be left for him to sort after school? Explain the error in Armand’s thinking. Use the manipulatives to support your reasoning.

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[B] Use manipulatives to solve this problem. Mark ate half of the candies in a bag. Leila ate 2/3 of what was left. Now there are 11 candies in the bag. How many were in the bag at the start? [C] Bill’s Snow Plow can plow the snow off the school’s parking lot in 4 hours. Jane’s plowing company can plow the same parking lot in just 3 hours. How long would it take Bill and Jane to plow the school’s parking lot together? Think of the math content involved with this problem Think of some “Before” activities that could be used. What would the debrief look/sound like in the classroom after the task was complete?

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Teaching Through Problem Solving [D] Mrs. Get Fit teaches Math and Phys-ed. To incorporate Math into the Phys-ed class, she divided the class into eight groups. There are three students in each group. The first person in the group runs ¼ of a lap of the track, the second person runs 1/6 of the track, and the third person runs 1/3 of a lap of the track. How many laps of the track are run in total by all eight teams combined?

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